Special Right Triangles

Hello guys :) This is chitwan and today's short blog is going to be about Special Right Triangles.

A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45–45–90. This is called an "angle-based" right triangle.

"Angle-based" special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or π/2 radians, is equal to the sum of the other two angles.

The side lengths are generally deduced from the basis of the unit or other geometric methods. This approach may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, and 60°.

Special triangles are used to aid in calculating common trigonometric functions, as below:

Degrees	Radians	sin	cos	tan
0	0
30
45
60
90

Let's look at the first of the special angle triangles, and this will hopefully become clear. Take a right angle triangle with two 45 degree angles, and with sides of 1 unit length. By the Theorem of Pythagoras, the hypotenuse of this triangle is of length √2. This is what this triangle looks like:

So then, from these values and SOHCAHTOA, you can obtain the trig values for this special angle of 45 degrees. You can see that:

Sin(45) = 1/√2
Cos(45) = 1/√2 
Tan(45) = 1

Don't worry if you can't remember these exact ratios... the simplest thing to remember is how to construct the special angle triangle... which is as easy as remembering a right angle triangle with a 45 degree angle and 2 sides of length 1... 

The second of the special angle triangles, which represents the rest of the special angles to remember, is slightly more complex, but still straightforward. Take a right angle triangle with angles of 30, 60, and 90 degrees. The simplest lengths of the sides of this triangle are 1, 2, √3 (with 2 being the longest side, the hypotenuse). This 30 60 90 triangle looks like this:

So then, using the these values, you can obtain the trig values for these special angles as well:

Sin(30) = 1/2
Cos(30) = √3/2
Tan(30) = 1/√3

Sin(60) = √3/2
Cos(60) = 1/2
Tan(60) = √3/1 = √3

And if you still don't get this or are dumb like me just go and look at the notes we did. Believe me they are much more simple and easier to understand :)

Here are some links that can help you with special right triangles: :p

http://www.youtube.com/watch?v=MHvHyv0aubw

http://www.youtube.com/watch?v=6Cb-XzSMXo4